Unconstrained integer quadratic programming

Question

What are the approaches for finding the global optimum of an unconstrained quadratic optimization problem where all variables are integers? When the integrality constraint is relaxed, the problem is convex. The variables can be binary as well.

Answer 1

For unconstrained integer convex optimization, you can solve the continuous relaxation by any method, and an integer optimal solution is obtained by rounding each component up or down. Explicitly, if the continuous optimal solution is $(x_1^*, \dots,x_n^*)$, an integer optimal solution is among the $$\prod_{j=1}^n (\lceil x_j^* \rceil - \lfloor x_j^* \rfloor + 1)\le 2^n$$ points of the form $(y_1^*, \dots, y_n^*)$, where $y_j^* \in \{\lfloor x_j^* \rfloor, \lceil x_j^* \rceil\}$.